few_shots: Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001097

7
options: Solve systems of equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Understand the connections between proportional relationships, lines, and linear equations.

Analyze patterns and relationships.

Perform arithmetic operations on polynomials.

Reason about and solve one-variable equations and inequalities.

Represent and solve problems involving multiplication and division.

Write expressions in equivalent forms to solve problems.

Write and interpret numerical expressions.

Use polynomial identities to solve problems.

Add and subtract within 20.

Solve equations and inequalities in one variable.

Understand solving equations as a process of reasoning and explain the reasoning.

Gain familiarity with factors and multiples.

Multiply and divide within 100.

Rewrite rational expressions.

Understand and apply properties of operations and the relationship between addition and subtraction.

Understand the relationship between zeros and factors of polynomials.

Represent and analyze quantitative relationships between dependent and independent variables.

Create equations that describe numbers or relationships

Use the four operations with whole numbers to solve problems.

Use properties of operations to generate equivalent expressions.

Represent and solve equations and inequalities graphically.

Generate and analyze patterns.

Interpret the structure of expressions.

Understand properties of multiplication and the relationship between multiplication and division.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Work with equal groups of objects to gain foundations for multiplication.

Work with addition and subtraction equations.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Work with radicals and integer exponents.

Represent and solve problems involving addition and subtraction.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.
problem_activity: Activity
This activity is a culmination of all the work students have done with angles in this unit. With less support than in previous activities, students come up with equations that represent the relationships between angles in a figure. Then, students solve their equation to find each unknown angle measure.
Launch
Encourage students to write an equation for each problem. Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge by asking students to start by looking for any vertical, complementary and supplementary angles. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Find the unknown angle measures. Show your thinking. Organize it so it can be followed by others.
[IMG]
[IMG]
Lines
\(\ell\)
and
\(m\)
are perpendicular.
[IMG]
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The diagram contains three squares. Three additional segments have been drawn that connect corners of the squares. We want to find the exact value of
\(a+b+c\)
.
[IMG]
Use a protractor to measure the three angles. Use your measurements to conjecture about the value of
\(a+b+c\)
.
Find the exact value of
\(a+b+c\)
by reasoning about the diagram.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students struggle to see the angle relationships in the figures, prompt them to look for any angles that are vertical, complementary, or supplementary to get them started.
Activity Synthesis
The goal of this discussion is for students to see different equations that can be used to represent and solve for the same unknown angle measures.
Select students to share their answers to each problem. Consider asking some of the following questions:
“Did anyone use a different equation for this same problem? If so, did you get the same answer?”
“Were any of the questions harder than others? Why?”
“Were there any questions you used a strategy that was new to you?”
Speaking: MLR8 Discussion Supports.
When selected students share how they calculated the angle measurements, invite other students to challenge an idea, elaborate on an idea, or clarify the idea using improved mathematical language. Encourage students to demonstrate central concepts of the angle relationships (e.g., complementary, supplementary, vertical) multi-modally by explaining their reasoning using images of the angles as well as gestures. This will support student understanding about how to write equations that represent the relationships between angles in a figure.
Design Principle(s): Optimize output (for explanation); Support sense-making


correct_option_index: 2